Subgroups — C₂³×#14556125 #32904139

Order 18176.

24030 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₂₈₄	C₂	1	6	yes	maximal, complemented
2	C₂²×#14556125	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄²×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵×D₇₁	C₂²	1	4	yes	—
5	C₂³×D₂₈₄	—	2	96	no	—
6	C₂³×D₂₈₄	C₂²	1	168	yes	complemented
7	C₂×#14556125	C₂²	1	448	yes	direct product factor
8	C₂⁴×C₄×C₇₁	C₂²	1	3	yes	—
9	C₂²×C₄²×C₇₁	C₂²	1	28	yes	complemented
10	C₂⁴×D₇₁	—	2	96	no	—
11	C₂⁴×D₇₁	C₂³	1	56	yes	—
12	C₂²×D₂₈₄	—	2	1344	no	—
13	C₂²×D₂₈₄	C₂³	1	672	yes	complemented
14	#14556125	C₂³	1	512	yes	direct product factor
15	C₂⁵×C₇₁	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₇₁	D₄	1	48	yes	complemented
17	C₂³×C₄×C₇₁	C₂³	1	42	yes	—
18	C₂×C₄²×C₇₁	C₂³	1	112	yes	complemented
19	C₂³×D₇₁	—	2	672	no	—
20	C₂³×D₇₁	—	4	256	no	—
21	C₂³×D₇₁	C₂⁴	1	112	yes	—
22	C₂×D₂₈₄	—	2	2688	no	—
23	C₂×D₂₈₄	C₂⁴	1	384	yes	complemented
24	C₂⁴×C₇₁	C₂×D₄	1	24	yes	—
25	C₂⁴×C₇₁	C₂⁴	1	7	yes	—
26	C₂²×C₄×C₇₁	C₂×D₄	1	336	yes	complemented
27	C₂²×C₄×C₇₁	C₂⁴	1	84	yes	—
28	C₄²×C₇₁	C₂⁴	1	64	yes	complemented
29	C₂³×C₇₁	#127	1	64	yes	complemented
30	C₂³×C₇₁	C₂²×D₄	1	84	yes	—
31	C₂³×C₇₁	C₂⁵	1	7	yes	—
32	C₂²×D₇₁	—	2	672	no	—
33	C₂²×D₇₁	—	4	896	no	—
34	C₂²×D₇₁	C₂⁵	1	32	yes	—
35	C₂×C₄×C₇₁	C₂²×D₄	1	336	yes	complemented
36	C₂×C₄×C₇₁	C₂⁵	1	24	yes	—
37	D₂₈₄	—	2	768	no	—
38	C₂²×C₇₁	C₂×#127	1	112	yes	complemented
39	C₂²×C₇₁	C₂³×D₄	1	42	yes	—
40	C₂²×C₇₁	C₂⁶	1	1	yes	characteristic
41	C₂×D₇₁	—	2	96	no	—
42	C₂×D₇₁	—	4	448	no	—
43	C₄×C₇₁	C₂³×D₄	1	48	yes	complemented
44	C₂³×#127	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	C₂²×#127	1	28	yes	complemented
46	C₂×C₇₁	C₂⁴×D₄	1	3	yes	—
47	D₇₁	—	4	32	no	—
48	C₂⁴×D₄	—	71	6	no	—
49	C₂²×#127	—	71	56	no	—
50	C₂³×C₄²	D₇₁	1	1	yes	characteristic, complemented
51	C₇₁	C₂³×#127	1	1	yes	characteristic, complemented, Sylow
52	C₂⁶	—	71	4	no	—
53	C₂³×D₄	—	71	168	no	—
54	C₂³×D₄	—	142	96	no	—
55	C₂⁴×C₄	C₂×D₇₁	1	3	yes	—
56	C₂×#127	—	71	448	no	—
57	C₂²×C₄²	C₂×D₇₁	1	28	yes	complemented
58	C₂⁵	—	71	56	no	—
59	C₂⁵	—	142	96	no	—
60	C₂⁵	C₂²×D₇₁	1	1	yes	characteristic
61	C₂²×D₄	—	71	672	no	—
62	C₂²×D₄	—	142	1344	no	—
63	C₂³×C₄	D₂₈₄	1	48	yes	complemented
64	C₂³×C₄	C₂²×D₇₁	1	42	yes	—
65	#127	—	71	512	no	—
66	C₂×C₄²	C₂²×D₇₁	1	112	yes	complemented
67	C₂⁴	—	284	256	no	—
68	C₂⁴	—	71	112	no	—
69	C₂⁴	—	142	672	no	—
70	C₂⁴	C₂×D₂₈₄	1	24	yes	—
71	C₂⁴	C₂³×D₇₁	1	7	yes	—
72	C₂×D₄	—	71	384	no	—
73	C₂×D₄	—	142	2688	no	—
74	C₂²×C₄	C₂×D₂₈₄	1	336	yes	complemented
75	C₂²×C₄	C₂³×D₇₁	1	84	yes	—
76	C₄²	C₂³×D₇₁	1	64	yes	complemented
77	C₂³	—	284	896	no	—
78	C₂³	—	71	32	no	—
79	C₂³	—	142	672	no	—
80	C₂³	#14556125	1	64	yes	direct product factor
81	C₂³	C₂²×D₂₈₄	1	84	yes	—
82	C₂³	C₂⁴×D₇₁	1	7	yes	—
83	D₄	—	142	768	no	—
84	C₂×C₄	C₂²×D₂₈₄	1	336	yes	complemented
85	C₂×C₄	C₂⁴×D₇₁	1	24	yes	—
86	C₂²	—	284	448	no	—
87	C₂²	—	142	96	no	—
88	C₂²	C₂×#14556125	1	112	yes	direct product factor
89	C₂²	C₂³×D₂₈₄	1	42	yes	—
90	C₂²	C₂⁵×D₇₁	1	1	yes	characteristic
91	C₄	C₂³×D₂₈₄	1	48	yes	complemented
92	C₂	—	284	32	no	—
93	C₂	C₂²×#14556125	1	28	yes	direct product factor
94	C₂	C₂⁴×D₂₈₄	1	3	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₇₁
commutator, cosocle	C₂²×C₇₁	C₂⁶
Frattini	C₂²	C₂⁵×D₇₁
Fitting	C₂³×C₄²×C₇₁	C₂
socle	C₂⁵×C₇₁	C₂³
hypercenter	C₂³×C₄²	D₇₁
hypocenter	C₇₁	C₂³×#127
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂⁴×D₂₈₄	C₂	semidirect
2. C₂²×#14556125	C₂	direct (N × H)
3. C₂³×C₄²×C₇₁	C₂	semidirect
6. C₂³×D₂₈₄	C₂²	semidirect
7. C₂×#14556125	C₂²	direct (N × H)
9. C₂²×C₄²×C₇₁	C₂²	semidirect
13. C₂²×D₂₈₄	C₂³	semidirect
14. #14556125	C₂³	direct (N × H)
16. C₂³×C₄×C₇₁	D₄	semidirect
18. C₂×C₄²×C₇₁	C₂³	semidirect
23. C₂×D₂₈₄	C₂⁴	semidirect
26. C₂²×C₄×C₇₁	C₂×D₄	semidirect
28. C₄²×C₇₁	C₂⁴	semidirect
29. C₂³×C₇₁	#127	semidirect
35. C₂×C₄×C₇₁	C₂²×D₄	semidirect
38. C₂²×C₇₁	C₂×#127	semidirect
43. C₄×C₇₁	C₂³×D₄	semidirect
45. C₂×C₇₁	C₂²×#127	semidirect
50. C₂³×C₄²	D₇₁	semidirect
51. C₇₁	C₂³×#127	semidirect
57. C₂²×C₄²	C₂×D₇₁	semidirect
63. C₂³×C₄	D₂₈₄	semidirect
66. C₂×C₄²	C₂²×D₇₁	semidirect
74. C₂²×C₄	C₂×D₂₈₄	semidirect
76. C₄²	C₂³×D₇₁	semidirect
80. C₂³	#14556125	direct (N × H)
84. C₂×C₄	C₂²×D₂₈₄	semidirect
88. C₂²	C₂×#14556125	direct (N × H)
91. C₄	C₂³×D₂₈₄	semidirect
93. C₂	C₂²×#14556125	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×D₂₈₄	2
C₂²×#14556125	2
C₂³×C₄²×C₇₁	2
C₂³×#127	71

Subgroups — C23×#14556125 #32904139

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556125 #32904139